Isotherm from dynamic model

In this paper we present results for a series of PEO fractions physically adsorbed on per-deutero polystyrene latex (PSL) in the plateau region of the adsorption isotherm. Hydro-dynamic and adsorption measurements have also been made on this system. Using a porous layer theory developed recently by Cohen Stuart (10) we have calculated the hydrodynamic thickness of these adsorbed polymers directly from the experimental density profiles. The results are then compared with model calculations based on density profiles obtained from the Scheutjens and Fleer (SF) layer model of polymer adsorption (11). 

The effects of bubble size and excess air rate on the transient average bed temperature are illustrated in Figure 5. The effect of bubble size is almost negligible under stable operating conditions, while the effect of excess air has a strong influence on the temperature change. It can be seen in both Figures 4 and 5 that the bed reaches a steady state at about 2000 s after initiation of the operation. This value is very different from the value, 200 s, obtained based on an isothermal dynamic model (10). 

Rathousky and Hlavacek (1981) presented two mathematical models to illustrate the fact that the influence of adsorbed species on the rate of an isothermal catalytic reaction may lead to a complex dynamic pattern including multiplicity of steady states and oscillatory states. Multiple oscillations and horatian behavior can not be calculated from the models. Rathousky and Hlavacek (1982) studied CO oxidation on Pt/Al203 catalyst and observed changes in oscillations due to the variations in inlet temperature. For a narrow range they observed horatian behavior. Experiments show that interaction of two oscillatory processes cause horatian behavior. 

Figure 7 Example of adsorption isotherms calculated from the dynamic model (illustrated in the insert) assuming (small) spontaneous desorption. The isotherms are drawn in two different scales and for two different relative magnitudes of the desorption terms.

Yu et al. (2011) studied rheology and phase separation of polymer blends with weak dynamic asymmetry ((poly(Me methacrylate)/poly(styrene-co-maleic anhydride)). They showed that the failure of methods, such as the time-temperature superposition principle in isothermal experiments or the deviation of the storage modulus from the apparent extrapolation of modulus in the miscible regime in non-isothermal tests, to predict the binodal temperature is not always applicable in systems with weak dynamic asymmetry. Therefore, they proposed a rheological model, which is an integration of the double reptation model and the selfconcentration model to describe the linear viscoelasticity of miscible blends. Then, the deviatirMi of experimental data from the model predictions for miscible... 

Consider the dynamics of an isothermal CSTR followed by a simple (single stage) separating unit (e.g., an extractor, crystallizer, or a settler). The reaction is reversible, A B, and the effluent stream from the separator, which is rich in unreacted material, is recycled. It is assumed that the reaction rate is first order, the equilibrium relationship for the separator is linear, and the rate of mass transfer between the phases in the separator could be written in terms of mass transfer coefficient and a linear driving force. Making certain that you define all your terms carefully, show that the dynamic model for the plant can be put into the following form ... 

In this chapter, tubular chemical reactors will be analyzed. Reaction mechanisms considered are first order in the components. As will be shown, at isothermal conditions reactor dynamics can be described by partial dififerential equations for the component balances. If non-isothermal conditions exist, the energy balance would also be a partial differential equation. Dynamics models are considerably different from the models for ideally stirred chemical reactors under similar operating conditions. 

When the two liquid phases are in relative motion, the mass transfer coefficients in eidrer phase must be related to die dynamical properties of the liquids. The boundary layer thicknesses are related to the Reynolds number, and the diffusive Uansfer to the Schmidt number. Another complication is that such a boundaty cannot in many circumstances be regarded as a simple planar interface, but eddies of material are U ansported to the interface from the bulk of each liquid which change the concenuation profile normal to the interface. In the simple isothermal model there is no need to take account of this fact, but in most indusuial chcumstances the two liquids are not in an isothermal system, but in one in which there is a temperature gradient. The simple stationary mass U ansfer model must therefore be replaced by an eddy mass U ansfer which takes account of this surface replenishment. 

Dynamic simulations for an isothermal, continuous, well-mixed tank reactor start-up were compared to experimental moments of the polymer distribution, reactant concentrations, population density distributions and media viscosity. The model devloped from steady-state data correlates with experimental, transient observations. Initially the reactor was void of initiator and polymer. 

Figure 15. Isotherms of internal mobilities in alkali-alkaline earth nitrate mixtures. The mobility of the alkali ion is always greater than that of the alkaline earth ion. (Reprinted from T. Koura, H. Matsuura, and I. Okada, "A Dynamic Dissociation Model for Internal Mobilities in Molten Alkali and Alkaline Earth Nitrate Mixtures,"/ Mol. Liq. 73-75 195, Fig. 4, Copyright 1997 with permission from Elsevier Science.)...

The dynamic surface tension of a monolayer may be defined as the response of a film in an initial state of static quasi-equilibrium to a sudden change in surface area. If the area of the film-covered interface is altered at a rapid rate, the monolayer may not readjust to its original conformation quickly enough to maintain the quasi-equilibrium surface pressure. It is for this reason that properly reported II/A isotherms for most monolayers are repeated at several compression/expansion rates. The reasons for this lag in equilibration time are complex combinations of shear and dilational viscosities, elasticity, and isothermal compressibility (Manheimer and Schechter, 1970 Margoni, 1871 Lucassen-Reynders et al., 1974). Furthermore, consideration of dynamic surface tension in insoluble monolayers assumes that the monolayer is indeed insoluble and stable throughout the perturbation if not, a myriad of contributions from monolayer collapse to monomer dissolution may complicate the situation further. Although theoretical models of dynamic surface tension effects have been presented, there have been very few attempts at experimental investigation of these time-dependent phenomena in spread monolayer films. 

The difference between the static or equilibrium and dynamic surface tension is often observed in the compression/expansion hysteresis present in most monolayer Yl/A isotherms (Fig. 8). In such cases, the compression isotherm is not coincident with the expansion one. For an insoluble monolayer, hysteresis may result from very rapid compression, collapse of the film to a surfactant bulk phase during compression, or compression of the film through a first or second order monolayer phase transition. In addition, any combination of these effects may be responsible for the observed hysteresis. Perhaps understandably, there has been no firm quantitative model for time-dependent relaxation effects in monolayers. However, if the basic monolayer properties such as ESP, stability limit, and composition are known, a qualitative description of the dynamic surface tension, or hysteresis, may be obtained. 

In conclusion, the maximum adsorption capacity should be measured in fixed-bed experiments under dynamic conditions, and if models are applicable, diffusion coefficients should be also determined in fixed-bed apparatus. Due to the fact that the equilibrium isotherms require extended data series and thus are time-consuming experiments, the latter are quite difficult to be conducted in fixed-bed reactors and from this point of view, it is more practical to evaluated equilibrium isotherms in batch reactor systems. Then, it is known that when applying fixed-bed models using an equilibrium isotherm obtained in batch-type experiments, the equilibrium discrepancy (if it exists) can be compensated by a different estimate for the solid diffusion coefficient (Inglezakis and Grigoropoulu, 2003 Weber and Wang, 1987). 

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